Mathematical Methods for Physics and Engineering : A Comprehensive Guide

NUMERICAL METHODS

appreciate how this would apply in (say) a computer program for a 1000-variable

case, perhaps with unforseeable zeros or very small numbers appearing on the

leading diagonal.

Solve the simultaneous equations

(a) x 1 +6x 2 − 4 x 3 =8,

(b) 3 x 1 − 20 x 2 +x 3 =12,

(c) −x 1 +3x 2 +5x 3 =3.

(27.22)

Firstly, we interchange rows (a) and (b) to bring the term 3x 1 onto the leading diagonal. In
the following, we label the important equations (I), (II), (III), and the others alphabetically.
A general (i.e. variable) label will be denoted byj.

(I) 3 x 1 − 20 x 2 +x 3 =12,

(d) x 1 +6x 2 − 4 x 3 =8,

(e) −x 1 +3x 2 +5x 3 =3.

For (j) = (d) and (e), replace row (j)by

row (j)−

aj 1

3

×row (I),

whereaj 1 is the coefficient ofx 1 in row (j), to give the two equations

(II)

(

6+^203

)

x 2 +

(

− 4 −^13

)

x 3 =8−^123 ,

(f)

(

3 −^203

)

x 2 +

(

5+^13

)

x 3 =3+^123.

Now|6+^203 |>| 3 −^203 |and so no interchange is required before the next elimination. To
eliminatex 2 ,replacerow(f)by

row (f)−

(

−^113

)

38

3

×row (II).

This gives

(III)

[ 16

3 +

11

38 ×

(−13)

3

]

x 3 =7+^1138 × 4.

Collecting together and tidying up the final equations, we have

(I) 3x 1 − 20 x 2 +x 3 =12,

(II) 38 x 2 − 13 x 3 =12,

(III) x 3 =2.

Starting with (III) and working backwards, it is now a simple matter to obtain

x 1 =10,x 2 =1,x 3 =2.

27.3.2 Gauss–Seidel iteration

In the example considered in the previous subsection an explicit way of solving

a set of simultaneous equations was given, the accuracy obtainable being limited

only by the rounding errors in the calculating facilities available, and the calcula-

tion was planned to minimise these. However, in some situations it may be that

only an approximate solution is needed. If, for a large number of variables, this is